This application claims benefit of Japanese Application No. 2000-396780 filed in Japan on Dec. 27, 2000, the contents of which are incorporated by this reference.
1. Field of the Invention
The present invention relates to an imaging simulation method and an imaging simulation system using the same, and also relates to a recording medium programmed with the imaging simulation method. More particularly, the present invention relates to a method of simulating imaging of an observation object performed by an image-forming optical system, and also relates to a simulation system using the imaging simulation method. Further, the present invention relates to a recording medium containing an imaging simulation program for instructing a computer to execute the imaging simulation method.
2. Discussion of Related Art
As methods of calculating imaging, including illuminating light, in microscopes, steppers, etc., those based on imaging theory according to Fourier optics are publicly known as shown, for example, in M. Born and E. Wolf, xe2x80x9cPrinciples of Optics, Sixth Editionxe2x80x9d, Pergamon Press (1980), Chapter X, and Tokuhisa Ito, xe2x80x9cOptics for Steppers (1) to (4)xe2x80x9d, Optical Technology Contact, Vol. 27 (1989), pp. 762-771, and Vol. 28 (1990), pp. 59-67, 108-119, and 165-175. For example, Kxc3x6hler illumination employed most frequently in microscopes and steppers uses an imaging system as shown in FIG. 12. As illustrated in the figure, the imaging system comprises an illuminating optical system including an effective source, an object, an image-forming optical system, and an image plane. The effective source is placed at the position of the pupil of the illuminating optical system or at a position conjugate to the pupil position. The pupil position of the illuminating optical system is conjugate to the pupil position of the image-forming optical system. The object position and the image plane are also conjugate to each other. In imaging calculation based on Fourier optics, the image intensity distribution I in the image plane is calculated from the following equation on the assumption that the intensity distribution S of the effective source, the transmission function a of the object and the pupil function P of the image-forming optical system can be expressed in terms of spatially two-dimensional scalar functions:
I(xcexd)=∫∫FS(u1xe2x88x92u2)a(u1)a*(u2)xc3x97FP(xcexdxe2x88x92u1)FP*(xcexdxe2x88x92u2)du1du2xe2x80x83xe2x80x83(1) 
where u1 and u2 are position coordinates in the object plane; xcexd is a position coordinate in the image plane; and FS and FP represent the Fourier transform of S and the Fourier transform of P, respectively.
Thus, in the imaging calculation based on Fourier optics, the image intensity distribution can be obtained by a simple integration using a Fourier transform of a scalar function determined by each constituent element of the optical system and the transmission function of the object. Further, this theory is convenient for the discussion of the spatial frequency characteristics of imaging. Therefore, it is widely used as imaging theory for microscopes, steppers, etc.
With the imaging calculation method based on Fourier optics, however, the object is expressed as a spatially two-dimensional scalar function. Therefore, it is impossible to accurately express the imaging of an object having a three-dimensional structure for which the thickness cannot be ignored. That is, an object with a thickness varies in transmittance according not only to the position where light rays pass but also to the inclination of the rays.
Under these circumstances, attempts have been made to perform imaging calculation for microscopes, steppers, etc. by a method wherein only the surrounding of the object is subjected to the electric field calculation in the three-dimensional space containing the object, and the remaining part is subjected to the conventional imaging calculation based on Fourier optics, as stated, for example, in G. L. Wojcik, et. al., xe2x80x9cNumerical Simulation of Thick Line Width Measurements by Reflected Lightxe2x80x9d, Proceeding SPIE, Vol. 1464 (1991), pp. 187-203, and G. Wojcik, et. al., xe2x80x9cSome Image Modeling Issues for I-line, 5xc3x97 phase shifting masksxe2x80x9d, Proceeding SPIE, Vol. 2197, pp. 455-465.
In the calculation of the electric field distribution in the three-dimensional space surrounding the object, the finite element method and the FDTD method [Finite Difference Time Domain method; e.g. K. Kunz and R. J. Luebbers, xe2x80x9cThe Finite Difference Time Domain Method for Electromagneticsxe2x80x9d, CRC Press (1993)] are generally used. An example of the FDTD method is shown in FIG. 9. In these methods, the space is divided into small cells, and the elements of the electromagnetic field are given as variables for each cell, thereby numerically obtaining a solution that satisfies the Maxwell""s equations with respect to the incident electromagnetic field. Accordingly, the solution obtained by these methods is a three-dimensional vector distribution in the three-dimensional space of a scattered electromagnetic field resulting from the incident electromagnetic field or the entire electromagnetic field as the sum total of these electromagnetic fields.
It is publicly known that the electromagnetic field entering the three-dimensional space surrounding the object can be determined by performing the Huygens-Fresnel diffraction integral on the intensity distribution S of the effective source at the pupil position of the illuminating optical system, as shown, for example, in M. Born and E. Wolf, xe2x80x9cprinciples of Optics, Sixth Editionxe2x80x9d, Pergamon Press (1980), Chapter VIII. It is also publicly known that when the NA (Numerical Aperture) of illuminating light is large, it is necessary to use a vector diffraction calculation such as that shown in B. Richards and E. Wolf. xe2x80x9cElectromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic systemxe2x80x9d, Pro. Roy. Soc. A, Vol. 253, pp. 358-379. Once the incident electromagnetic field and the shape of the object are determined, it is possible to calculate a scattered electromagnetic field resulting from the incident electromagnetic field scattered by the object or the entire electromagnetic field as the sum total of the incident electromagnetic field and the scattered electromagnetic field.
Meanwhile, in the process of calculating imaging performed by the image-forming optical system from the obtained three-dimensional vector distribution of the electromagnetic field in the three-dimensional space surrounding the object, a method using Fourier optics as it is has heretofore been employed in general. Basically, Fourier optics can handle only spatially two-dimensional scalar functions, as has been stated above. Accordingly, it is necessary to convert the three-dimensional vector quantity of the electromagnetic field into a scalar quantity. Therefore, the conventional practice is to adopt a 2.5-dimensional model for the object and the incident electromagnetic field.
The term xe2x80x9c2.5-dimensional modelxe2x80x9d means a model in which, as shown in FIGS. 13(a) and 13(b), the object and the electromagnetic field occupy a three-dimensional space (x, y, z), but both of them are in translational symmetry with respect to a one-dimensional direction (y). In the 2.5-dimensional model, the entire electromagnetic field separates independently into a TE mode [FIG. 13(a)] in which the electric field component is parallel to the y-direction, and a TM mode [FIG. 13(b)] in which the magnetic field component is parallel to the y-direction. The electromagnetic field in each of the two modes can be expressed in terms of a scalar quantity because the polarization direction is uniform. Accordingly, it can be applied to Fourier optics. More specifically, in each of the TM and TE modes, the electric field distribution in the focal plane of the image-forming optical system is obtained and convolved with the PSF (Point Spread Function) of the image-forming optical system. Alternatively, the electric field distribution is Fourier-transformed and multiplied by the pupil function of the image-forming optical system and then inversely Fourier-transformed.
Incidentally, the above-described conventional method involves the following disadvantages:
As the NA of illuminating light increases, it becomes impossible to perform an accurate electromagnetic field calculation with the 2.5-dimensional model. The reason for this is as follows. Because illuminating light is incident on the object in the shape of a cone, there are a large number of rays tilted to a considerable extent with respect to the xz-plane. Therefore, it becomes inappropriate to approximate the entire electromagnetic field with only the TE and TM modes. Further, it is impossible to calculate the imaging of an object that is not conformable to the 2.5-dimensional configuration.
In addition, it is impossible with the above-described method to calculate imaging when the image-forming optical system is focused to the inside of the object. The reason for this is that the electric field actually existing inside the object differs from the apparent electric field in the object as observed from the outside. Similarly, when the image-forming optical system is focused to the bottom of a narrow and deep hole in the object, for example, imaging cannot accurately be calculated because the influence on the electromagnetic field of the wall surface of the hole closer to the image-forming optical system than the focal position cannot be ignored.
The present invention was made in view of the above-described problems with the prior art.
Accordingly, an object of the present invention is to provide a method capable of accurate imaging calculation (simulation) even when the NA of illuminating light is large, or when the object under observation is not conformable to the 2.5-dimensional configuration, or when the image-forming optical system is focused to the inside of the object. The present invention also provides a system using the imaging simulation method and a recording medium containing an imaging simulation program for instructing a computer to execute the imaging simulation method.
To attain the above-described object, the present invention provides an imaging simulation method including a first step of calculating the electric field distribution of illuminating light entering a three-dimensional space containing an observation object. The imaging simulation method further includes a second step of calculating the electric field distribution in the three-dimensional space of scattered light generated by interaction between the illuminating light and the observation object, and a third step of calculating imaging of the scattered light performed by an image-forming optical system by using the Helmholtz-Kirchhoff integral using a characteristic Green""s function group of the image-forming optical system.
Still other objects and advantages of the invention will in part be obvious and will in part be apparent from the specification.
The invention accordingly comprises the features of construction, combinations of elements, and arrangement of parts which will be exemplified in the construction hereinafter set forth, and the scope of the invention will be indicated in the claims.